Voronoi diagram on a Riemannian surface
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Title 
Voronoi diagram on a Riemannian surface
<abstract language="fre"> On s'intéresse à la mosaïque de Voronoi engendrée par un processus ponctuel de Poisson homogène sur une surface riemannienne. Plus précisément, on montre un lien entre les caractèristiques moyennes et d'une cellule et la courbure gaussienne de la surface. On se concentre sur le nombre moyen de sommets en commançant par donner une formule exacte dans le cas d'une sphère. On généralise ensuite à une surface quelconque en donnant un développement asymptotique à grande intensité. La preuve repose principalement sur des résultats classiques de comparaison issus de la géométrie riemannienne. </abstract>

Alternative Title 
Mosaïques de Voronoi sur une surface riemannienne

Title of Series  
Part Number 
8

Number of Parts 
17

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

Content Metadata
Subject Area 
00:00
Complex (psychology)
Spherical trigonometry
Group action
Correspondence (mathematics)
Sphere
Mereology
Orbit
Explosion
Maxima and minima
Mathematics
Matrix (mathematics)
Positional notation
Cuboid
Diagram
Vertex (graph theory)
Pole (complex analysis)
Perimeter
Position operator
Area
Curvature
Link (knot theory)
Logical constant
Process (computing)
Theory of relativity
Point (geometry)
Moment (mathematics)
Uniform convergence
Thermal expansion
Measurement
Proof theory
Arithmetic mean
Right angle
Sinc function
Geometry
Point (geometry)
Statistics
Link (knot theory)
Characteristic polynomial
Student's ttest
2 (number)
Number
Frequency
Goodness of fit
Arithmetic mean
Wellformed formula
Radius
Surface
Expression
Counting
Volume (thermodynamics)
Sphere
Number
Computer animation
Vertex (graph theory)
Object (grammar)
06:04
Point (geometry)
Greatest element
State of matter
Direction (geometry)
Volume (thermodynamics)
Coordinate system
Weight
Variable (mathematics)
Number
Mathematics
Meeting/Interview
Wellformed formula
Vertex (graph theory)
Physical system
Point (geometry)
Expression
Moment (mathematics)
Volume (thermodynamics)
Thermal expansion
Total S.A.
Geodesic
Similarity (geometry)
Mathematics
Arithmetic mean
Wellformed formula
Computer animation
Marginal distribution
09:35
Axiom of choice
Point (geometry)
Curvature
Multiplication sign
Image resolution
Distance
Variable (mathematics)
Frequency
Maxima and minima
Mathematics
Jacobi method
Thermodynamisches System
Wellformed formula
Forest
Position operator
Fiber (mathematics)
Directed graph
Pairwise comparison
Curvature
Standard deviation
Theory of relativity
Scaling (geometry)
Sine
Closed set
Gender
Expression
Mathematical analysis
Volume (thermodynamics)
Incidence algebra
Functional (mathematics)
Sphere
Estimator
Mathematics
Computer animation
Dew point
Triangle
Theorem
13:08
Point (geometry)
Closed set
Prisoner's dilemma
Expression
Volume (thermodynamics)
Volume (thermodynamics)
Barrelled space
Thermal expansion
Infinity
Bilinear form
Number
Computer animation
Arithmetic mean
Chain
Theorem
Vertex (graph theory)
14:38
Geometry
Surface
Curvature
Link (knot theory)
Direction (geometry)
Characteristic polynomial
Area
Number
Manysorted logic
Arithmetic mean
Meeting/Interview
Hausdorff dimension
Vertex (graph theory)
Perimeter
Area
Curvature
Link (knot theory)
Concentric
Characteristic polynomial
Volume (thermodynamics)
Line (geometry)
Integraltafel
Perspective (visual)
Fisher information
Number
Arithmetic mean
Riemann curvature tensor
Scalar field
Hausdorff dimension
Negative number
Units of measurement
Perimeter
00:02
we have at best the government has said that the better the in good morning I'm going to talk in English and on full 1st of all I would like to thank the organisers for this opportunity to present the work an going to talk about voluntary diagram on the Romanians your face currently doing as a PhD student we get continue to flow and that in any case and the University of not having West so what's the main idea we drove on a diagram on face and we would like to recover the local geometry of this face from the statistics of the so this is required to show a link between the main characteristics of the monitors said such as area a number of vertices of perimeter as the look characteristics of this your face and murderous it would be as a Galician correction so the following 4 some notations are in the stock as would be Romanians face with his Romanian between the and it's only measure of G X induced by symmetric and fine would be of no major news person .period process which is his respect to this matrix and we would be interested buyers of location Eritrea Rwanda fixpoint X number of and to do this we will investigate the mean number of entities of the bombers says effects so the demanding to box in the 1st part we investigate for the simplest strips surfaced again find which is the sphere of constant ratcheting and I'm really give up and geometric proved to get an exact formula for the mean number of entities in this case an innocent but I really show to adapt this method with the geometry the tours of women in geometry but to get an expansion that I intensity for the mean number of entities on an arbitrary surface so let's start with the scare in the case of the spin of Constanta Richard Jane that mean does reduce with 1 of us were of cake out we have this formula still the minimum to see that we have a stake as a beginning which is Zerk and constant because we'll the shorelines of playing we have fixed falls in the number of entities and it is by to dance that goes to 0 and went on that list to ingenuity the first one is the father of 1 of them and that Anderson 1 wonders it decreases exponentially fast 1 on that list 2 engines so we can know that as if expression on the depends of the rescue jail on that this can be explained because virus the sphere we can vary as intensity so mines in the seventies after the same reason but for a fixed number of off vertices on all points on the stairs and the strong relies on the correspondence between the registers of the diagram and as the faces of complex body address and use is under a formula solarsystem views on you valuable for this year and seems we have we are interested for more generous face and we to introduce pull and moderate report to get this exact formula so the proof is divided in 4 steps the first one is since we have to come vertices we have to characterize it and are in fact agent that is just as the 2nd center of X and 2 points of the process despite boy is empty of all the points of the process Friday contains no other points of the prestigious fight and is these are characterized as the best use of so 2 counts as an entity of Simi just that to some of the the valve .period of fight and since on the stairs 3 points always defined to circumscribe borders we have to some of the 2 seconds when sold me this expression region blight as the went on magazine that from that to get at the moment it as to computers indolent agronomy going to do to change is a viable the first one is sinking as as we just use spherical coordinates so after switching the industry coordinates we this expression and we still have to compute the volume of the 2 seconds by boat to the evaluators interval so to do this we make several changes viable which is a blast at the continent that change viable which is like this so the idea is to replace a spherical coordinates of X 1 will by the spherical coordinates of the significance and disease and that defining the position of X 1 the next to on the Bhandari of the bordered by a sea 1 that so it thanks to users spherical trigonometry formula that we can have this relations on the right and we can compute Zyrtec objectives a change viable and so he is expression where I isn't 1 over all as the possible imposition of 6 won the next 2 on the diary and in this new coordinates we have exact formula for the volume of the 2 seconds when boards lands east and so we can after computing this gives expression just announced is the idea now to do this
06:05
thought the margin has to face so we take another Teresa face and that too find a way to add this method to Inc to get at the mean number of entities but insists it's quite unlikely that we could get an exact formula in the case of a more generous to face so we are I'm going to show up to and that this method but for 2 to get and that some to take expansion at an intensity so the steps that will be under the same we have to characterize the rest of the industry and not be so different as for this year then we have to apply it to make a snap for me and at this moment we show that only points close to achieve X 2 X No we contribute significantly in the expansion so we make a distinction between point near 4 picks near 2 x knowledge and far from and then we really need is that would change is a viable is the same as before so it would require us to define should indeed but our coordinates to replace the spherical coordinates we used on the steer and to make a blast the constant that Ginger viable as the same as before but the acting more generous to face we haven't as a trigger military formula to get the expression of this change revival so we haven't is an exact change of liability and there is an additional step To find as the volume of the show but in the case of the span after a similar change of viable and we had the exact formula for the volume of the 2 separate language that not the case of In the case of more face so we have to computers use a phone so let's go but that 1 to characterize notices of seats the same as before we have to some of us a pair of points of fight and to some of the statements by boat which is not unique and we have to apply the makers net formula but a full points fast from X noted we can see that as a 2nd fled we'll have a 2 large volume and always likeness on die in front of this volume we can show that it decreases exponentially fast when number ingenuity and it will be negligible full by Starr excellent we can show it is shown that only 1 circumscribed board will contribute 0 there would be too large and between contribute and immediately but for the UNIX systems by Boris uh we need similar changes a viable as for this year still the
08:52
first one is to replace a spherical coordinates for use in the case of the state and it is replaced by his originally but Germany but our coordinates are just on just 0 put our coordinates of the total bottom playing at X not but projected on the Sophies buys expansion concretely that orange fight if you start from X not an issue warnings the direction given by fighting for spot you get the points on the on your serve face and he does put up coordinates are fight and since we have to make a change of itinerary we have 2 computers are associated to the change and in fact
09:37
we haven't any information about the Jacoby St yeah but we have a combined there and the launch launched to and which is that it the version given a choice of bonded from Berlin from evolved by 2 stunned then there is the sole bonded from borrowing from abroad by 2 functions which I exactly the questioning functions on the sphere of most on the adjustment time data still since we are in the across the board around X noted and scenes as the Bush encourages is continuous we are going to apply sister and we smiled and indignant that close to Jefferts so we
10:19
can make this change of vitamins and in this expression so that the audience negligible contributions of the board I just program and for the unique was which contributes we have made these changes viable an approximate viable and no we have we still have to compute the volume of the circumscribed and to do this remains a symbol gender violence which is the best accounting that change but here we would like to replace the suspect is to put out coordinates of X 1 annexed to the resolution of the board coordinates of the and the findings of position of X 1 the next 2 on the Bhandari of the voted by 50 . 1 incident but we have not so relations between all this environment but we have we need and we have a another comparison residence which is the top of the bestseller the point analysis to compare a small triangles you on face is it is if there is a Gaussian Georgia wandered from below and formidable by 2 Constance pastoral harsh so that we can compare smart goes on as we is trial on is the status of most sports Tonkin adjustment and that and dead Is that way that let's take 2 segments small segments of tanks with a fixed and in the between the 2 segments and if you focus on the distance between the 2 extremes .period then the of a standardized Rogers the curvature of these smaller as a distance would be in between the 2 .period this can be explained because a 1 0 career choice not visually tend to get close faster so the distance between the 2 points is a smart it so it is there and we as we trigonometry formula on scale of Muslim judges Modern Tire and in that dark false modern that that used to Jeff it's not as before we can get this change of viable so we have a network estimation off is the relation between the valuables and so we can compute an approximate gently and which is under the same as before so we have expression I use exactly the same in the world as forest this year and in this will we still have to compute the volume of the circus fiber that is 2nd by a new coordinates in just over the border of Standards the and reduce are and this
13:09
volume but is given by the death get prisoners we share that uh the volume of small due to the board of use of it is just as the volume of the neck again offered spark the great invited them off their offer the bonding of the Camacho adds but In our case since we are in the close neighbor or what is the point of the Israeli close to their X not so if we replace jails lead by Jeff not this expressions of barrel remake is contained in the remaining of Thomas More so we can do this and we get this
13:51
expression but then and the isn't while it remains to compute the other is famous forms and it is a classically computed by leftist missiles and we can get as this expression for the Minamata disease so we have this expansion for I intensity so we recover 6 which should correspond to the kid and and we haven't dancers but down which goes to his you win on that restrictions which is exactly as the same as far as the spirit replacing chain Baggio fix and with the remaining dams after sold
14:40
just to finish before lunch as Ansa face a understood adoptable and if I we have shown a link between the mean number of entities and as the Gaussian Corretja I present in this resort in a positive direction facetoface but this really still available for service of negative curvature so at the expense of soul is is a resident of isn't going to thousands of homes in the bill explain as I mentioned earlier in June 28 channels on all the characteristics such as area and perimeter but they're not to talk about seasonal and a no and we are going on the dimension greater than 3 so we are interested on is a look at geometry of many fond of dimension breed of 3 and up to some of the guys we have to check that some weeks ago we have proved that the reasoning between them in number of 2 seasons cannot Camacho the inserting in following money for the dimensions and and know what we are interested by standing and other characteristics to get the courage of because as we would like to recover more informations about the local geometry of this twofaced and although lenses candidature and there is the soul of a very questions to duties in it this 1st so I stop here and thank you
16:32
message on adult locally produced so the radial the contemplation of this fall sort of the you at concentration is the largest of the CEO and head I under moments in particular are in in the last line and it is quite a difficult to compute a powerful violence even in the case of the oxygen case it is given by your some disgusting integral formula and it that it requires greatest country some volumes of Hunan what of borders and it is not treating the 2 contradictions In this case so I haven't yet this had the scandal with quality E. mess yet Anthony few